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Properties

Label 2-396-12.11-c1-0-15
Degree 22
Conductor 396396
Sign −0.887+0.460i-0.887 + 0.460i
Analytic cond. 3.162073.16207
Root an. cond. 1.778221.77822
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.929 − 1.06i)2-s + (−0.272 + 1.98i)4-s − 3.24i·5-s − 1.06i·7-s + (2.36 − 1.55i)8-s + (−3.45 + 3.01i)10-s + 11-s − 0.185·13-s + (−1.13 + 0.985i)14-s + (−3.85 − 1.07i)16-s − 1.21i·17-s − 0.402i·19-s + (6.42 + 0.884i)20-s + (−0.929 − 1.06i)22-s − 8.43·23-s + ⋯
L(s)  = 1  + (−0.657 − 0.753i)2-s + (−0.136 + 0.990i)4-s − 1.45i·5-s − 0.400i·7-s + (0.836 − 0.548i)8-s + (−1.09 + 0.953i)10-s + 0.301·11-s − 0.0514·13-s + (−0.302 + 0.263i)14-s + (−0.962 − 0.269i)16-s − 0.294i·17-s − 0.0923i·19-s + (1.43 + 0.197i)20-s + (−0.198 − 0.227i)22-s − 1.75·23-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((−0.887+0.460i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+1/2) L(s)=((−0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: −0.887+0.460i-0.887 + 0.460i
Analytic conductor: 3.162073.16207
Root analytic conductor: 1.778221.77822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ396(287,⋅)\chi_{396} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :1/2), −0.887+0.460i)(2,\ 396,\ (\ :1/2),\ -0.887 + 0.460i)

Particular Values

L(1)L(1) ≈\approx 0.189685−0.777082i0.189685 - 0.777082i
L(12)L(\frac12) ≈\approx 0.189685−0.777082i0.189685 - 0.777082i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.929+1.06i)T 1 + (0.929 + 1.06i)T
3 1 1
11 1−T 1 - T
good5 1+3.24iT−5T2 1 + 3.24iT - 5T^{2}
7 1+1.06iT−7T2 1 + 1.06iT - 7T^{2}
13 1+0.185T+13T2 1 + 0.185T + 13T^{2}
17 1+1.21iT−17T2 1 + 1.21iT - 17T^{2}
19 1+0.402iT−19T2 1 + 0.402iT - 19T^{2}
23 1+8.43T+23T2 1 + 8.43T + 23T^{2}
29 1+7.33iT−29T2 1 + 7.33iT - 29T^{2}
31 1+8.06iT−31T2 1 + 8.06iT - 31T^{2}
37 1+0.871T+37T2 1 + 0.871T + 37T^{2}
41 1+6.18iT−41T2 1 + 6.18iT - 41T^{2}
43 1−9.72iT−43T2 1 - 9.72iT - 43T^{2}
47 1+4.14T+47T2 1 + 4.14T + 47T^{2}
53 1−11.7iT−53T2 1 - 11.7iT - 53T^{2}
59 1+10.0T+59T2 1 + 10.0T + 59T^{2}
61 1−4.64T+61T2 1 - 4.64T + 61T^{2}
67 1+2.66iT−67T2 1 + 2.66iT - 67T^{2}
71 1−9.80T+71T2 1 - 9.80T + 71T^{2}
73 1−14.1T+73T2 1 - 14.1T + 73T^{2}
79 1+13.5iT−79T2 1 + 13.5iT - 79T^{2}
83 1−17.7T+83T2 1 - 17.7T + 83T^{2}
89 1−15.1iT−89T2 1 - 15.1iT - 89T^{2}
97 1−2.18T+97T2 1 - 2.18T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.90759973936180673486589523473, −9.750228927028808570937841647920, −9.309704985066995631479791163325, −8.216866151254820678598865274057, −7.66209248785769377492378944997, −6.09857040028372302510689411482, −4.64264298609389426526313846924, −3.88839786124687670639656945679, −2.09270257683690935699028806347, −0.65529544868567777771118110202, 2.03478821721680913253810186897, 3.57133310778741382314957027886, 5.21703740929479802338136022387, 6.33758362517378647095157571313, 6.88876810547278730511190813485, 7.913479830766678379650913108792, 8.823081027159327241873810506558, 9.942850435695832112522480957535, 10.52262979014230277668716904932, 11.36708926723157417799422536858

Graph of the ZZ-function along the critical line